LREtools
riccati
find solutions of Riccati Recurrence equations
Calling Sequence
Parameters
Description
Examples
riccati(problem)
problem
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problem statement or RESol
Attempts to solve Riccati recurrence equations using various substitutions.
A Riccati recurrence equation in y(k) is one of the form yk+1yk+Akyk+1+Bkyk=Ck where A(k), B(k), and C(k) are independent of y(k). If the equation is homogeneous (Ck=0), then we try the substitution xk=1yk, which makes the equation first order linear and if not, then we try yk=xk−Bkxk+1xk+1, which makes the equation second order linear. Finally, there is the substitution yk=xk+1−Ak+1xkxk, which makes the equation second order linear. If rsolve can solve these new equations, then we back-substitute to obtain solutions to the original problems.
If A(k) is undefined for some k, then a set of equations may be returned, giving values of y(k) for specific k as well as the general formula.
Since it calls rsolve, this procedure can be expensive; because of the back-substitution, the answers may be overly complicated.
See the help page for LREtools[REcreate] for the definition of the format of a problem.
withLREtools:
prob≔REcreateyk+1yk+2yk+1−3yk=1,yk,y0=12
prob≔RESolyk+1−3yk+2yk+1=1,yk,y0=12,INFO
riccatiprob
−5−5k5−−5+5k5+−5−5k+−5+5k2−5+5k+−5−5k
riccatiyk+1yk+2yk+1−kyk=0,yk,∅
charfcn0ky0
See Also
LREtools[REcreate]
RESol
rsolve
Download Help Document